On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.     

On groups factorized by two subgroups with Chernikov commutants

We establish results concerning the almost solvability and other properties of groups factorized by two subgroups with finite or Chernikov commutants.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

On cyclicity of commutants

In this note an example is given of a Hilbert space operator whose commutant is not finitely cyclic. The example settles in the negative a conjecture of D. A. Herrero.     

On the relative commutants of subfactors

Let be factors generated by a periodic tower of finite dimensional -algebras. We prove that for sufficiently large , is -isomorphic to a subalgebra of .

     

Finite alperin 2-groups with cyclic second commutants

An Alperin group is a group in which every 2-generated subgroup has a cyclic commutant. Previously, we constructed examples of finite Alperin 2-groups with second commutant isomorphic to Z ₂ or Z ₄. Here, it is proved that for any natural n, there exists a finite Alperin 2-group whose second commutant is isomorphic to Z _2n .     

On the theory of finite groups

This is the author's abstract of his dissertation in the competition for the academic degree of Doctor in the Physico-Mathematical Sciences. The dissertation was defended on the 21st of January, 1969 at the Academic Council of the Institute of Mathematics of the Academy of Sciences of the USSR. The official opponents were: S. N. Chernikov, Corresponding Member, Academy of Sciences of the USSR and Professor B. I. Plotkin, Doctor of Physico-Mathematical Sciences.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 347–360, September, 1969.     

On the equipartitions of finite groups

Summary It is proved that the p- groups of submaximal class do not admit equipartitions other than the atomic (Theorem 3.2). This result is then wed to give a complete classification of the groups of order p⁶ admitting nonatomic equipartitions (Theorem 5.1).     

On the supersolvability of finite groups

Archiv der Mathematik -     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.